Integrable systems arise in nonlinear processes and, both in their classical and quantum version, have many applications in various fields of mathematics and physics, which makes them a very active research area. In this thesis, the problem of integrability of multidimensional equations, especially in three dimensions (3D), is explored. We investigate systems of differential, differential-difference and discrete equations, which are studied via a novel approach that was developed over the last few years. This approach, is essentially a perturbation technique based on the so called method of dispersive deformations of hydrodynamic reductions . This method is used to classify a variety of differential equations, including soliton equations and scalar higher-order quasilinear PDEs. As part of this research, the method is extended to differential-difference equations and consequently to purely discrete equations. The passage to discrete equations is important, since, in the case of multidimensional systems, there exist very few integrability criteria. Complete lists of various classes of integrable equations in three dimensions are provided, as well as partial results related to the theory of dispersive shock waves. A new definition of integrability, based on hydrodynamic reductions, is used throughout, which is a natural analogue of the generalized hodograph transform in higher dimensions. The definition is also justified by the fact that Lax pairs the most well-known integrability criteria are given for all classification results obtained.

Thèse diffusée initialement dans le cadre d'un projet pilote des Presses de l'Université de Montréal/Centre d'édition numérique UdeM (1997-2008) avec l'autorisation de l'auteur.

Det finns många studier på hur främmande arter påverkar en arts återinförsel, men det finns fåsom har gjorts om infödda arters påverkan. Atlantlax (Salmo salar) kunde en gång i tidenmigrera från Vänern i Sverige till Klarälven på den norska sidan. Det kan den inte idag, menidag talas det om att återinföra laxen till Klarälven i Norge så att den kan utföra hela sinmigrationsrutt igen. I den älven har laxen en gång i tiden koexisterat med sina konkurrenterharr (Thymallus thymallus), och öring (Salmo trutta) men idag är frågan hur denkonkurrensen mellan dessa tre arter kan komma att påverka återinförseln av lax. För att få enbättre bild av hur utgången för en potentiell återintroduktion kan se ut, så var syftet meddenna undersökning att ta reda på mot vilken art dessa tre arter främst riktade sinaaggressionsbeteenden mot. I akvarieförsök med juvenila individer av dessa tre arter harundersökningarna tagit plats och aggressiva beteenden filmades när arterna var i olikakombinationer. Resultatet visade att harr var den mest aktiva av de tre arterna och lax denminst aktiva, samt att lax blev dominerad utav både harr och öring. Resultatet i denna studietyder på att en återintroduktion av lax kan innehålla svårigheter, men i naturen är ett habitatmer komplext än i ett akvarium vilket kan ge ett annat resultat som kanske gör enåterintroduktion mer framgångsrik. / There are many studies about how the effects from exotic species could affect a reintroductionof a species, but there are only a few that have been made about the effects from nativespecies. The Atlantic salmon (Salmo salar) could historically speaking, migrate from lakeVänern in Sweden to the River Klarälven in Norway. Today it cannot do this, but there is anongoing discussion about reintroducing the salmon to the Norwegian side of River Klarälven,so it once again can swim along its entire migration route. In that river, the salmon used tocoexist with its competitors, grayling (Thymallus thymallus) and trout (Salmo trutta), buttoday the question is how competition between these three species could affect thereintroduction of salmon. To get a better understanding of the outcome of a possiblereintroduction, the purpose of this study was to ascertain which species these three speciesmainly directed their agonistic behavior towards. The study took place in aquariums withjuvenile individuals of the three species. Agonistic behavior was filmed when the specieswere together in different combinations. The results show that grayling is the most active ofthe three species, while salmon is the least active, and salmon was dominated by both graylingand trout. The results from this study indicates that a reintroduction of salmon might befraught with problems, but in nature a habitat is more complex than in an aquarium, whichmeans that in nature the results could be different, which might make a reintroduction moresuccessful.

Short Pulse equation derived by Schafer and Wayne is a nonlinear partial differential equation that describes ultra short laser propagation in a dispersive optical medium such as optical fibers. Some properties of this equation e.g. traveling wave solution and its soliton structure and some of its conserved quantities were investigated. Conserved quantities were obtained by mass conservation law, lax pair method and transformation between Sine-Gordon and short pulse equation. As a result, loop soliton characteristic and six conserved quantities were found.

In this thesis, we investigate the integrability properties of some evolutionary type nonlinear equations in (1+1)-dimensions both with commutative and non-commutative variables. We construct the recursion operators, based on the Lax representation, for such equations. Finally, we question the notion of integrability for a certain one-component non-commutative equation. [We stress that calculations in this thesis are not original.]

Thèse diffusée initialement dans le cadre d'un projet pilote des Presses de l'Université de Montréal/Centre d'édition numérique UdeM (1997-2008) avec l'autorisation de l'auteur.

Techn. University, Diss., 2002--Braunschweig.

In this work we apply the method of hydrodynamic reductions to study the integrability of the class of second order quasilinear equations.

Rising temperatures, because of climate change, will have major consequences for the world's fish populations, including the Atlantic salmon Salmo salar L. The greatest temperature changes are expected to occur during winter, which will affect S. salar since the eggs are developing during this time of year. Several studies have shown that elevated temperatures during embryogenesis cause morphological changes, in S. Salar, that are shown in later life stages. Some of these studies indicate that eggs incubated at high temperatures should generate parr with deeper bodies. To investigate whether high temperatures during the egg stage cause changes in body shape, parr from normal and high temperature incubated eggs, referred to as as “cold” and “warm” fish, respectively, were examined. A box-truss of euclidean distances between 10 landmarks on the fish body and a discriminant function analysis (DFA) was used to determine which of the distances best discriminated between the two groups. The test showed a significant discrimination between cold and warm parr regarding body shape where cold parr had greater heads and warm parr were deeper over the tail region and showed greater distances between the pelvic fin and the front attachment of the anal fin. 67,3 % of all parr where correctly classified by the test. These results support those notions that indicates that the incubation temperature is important for the morphological development of S. salar although it does not support the hypothesis. / Stigande temperaturer, till följd av klimatförändringar kommer att få stora konsekvenser för jordens fiskpopulationer, däribland atlantlaxen Salmo. Salar L. Störst temperaturförändringar väntas ske vintertid, vilket kommer att påverka S. Salar som övervintrar sin rom. Flera studier har visat att förhöjda temperaturer under embryogenesen ger upphov till morfologiska förändringar hos S. Salar under senare livsstadier. Några studier pekar mot att högtemperaturinkuberad rom torde ge upphov till parr med djupare kroppsform. För att undersöka om höga temperaturer under romstadiet ger upphov till förändringar beträffande kroppsform undersöktes parr (kalla resp. varma) från normal-  respektive högtemperaturinkuberad rom. Med hjälp av ett rutnät / box-truss mättes avstånd mellan 10 landmärken på fiskkroppen och en discriminant function analysis DFA användes för att avgöra vilket av avstånden som bäst diskriminerade mellan de båda grupperna. Testet visade en signifikant diskriminering mellan kalla och varma parr beträffande kroppsform där kalla parr hade större huvudmått och varma parr hade djupare mått över stjärtspolen och längre mått mellan analfenans främre infästning och bukfenan.  67,3 % av all parr förutspåddes tillhöra rätt grupp. Dessa resultat är i linje med den litteratur som anger att inkuberingstemperaturen har betydelse för den morfologiska utvecklingen hos S. Salar däremot stödjer de inte den angivna hypotesen.

The subject of the submitted thesis „Low pressure water hammer“ is handling the problems of dynamic conduct of water in low pressures. In this work are developed problems of the effect of changing pressure on the speed of sound in liquid. The problem is solved in the time domain using the Lax - Wendroff method in the theoretical analysis. At the end the mathematical model solved using Lax - Wendroff method is confront with the experiment.